This chapter discusses aperiodic crystals. The structure and symmetry of quasiperiodic crystals can be described by embedding them into a higher-dimensional space as lattice periodic structures. ...
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This chapter discusses aperiodic crystals. The structure and symmetry of quasiperiodic crystals can be described by embedding them into a higher-dimensional space as lattice periodic structures. Their intersection with the physical space gives the real structure, shifting the physical space parallel to the original one gives another possible realization of the crystal with the same energy. The Fourier transform of the quasiperiodic structure, and its diffraction pattern, are projections of the corresponding quantities in higher dimensions. The symmetry groups of quasiperiodic structures are superspace groups, higher-dimensional space groups for which the point group can be decomposed into a component in physical space and one in the additional, internal space. The structure determination reduces to the determination of number and positions of atomic surfaces in the higher-dimensional unit cell, and that of the shape of the atomic surfaces.Less

DESCRIPTION AND SYMMETRY OF APERIODIC CRYSTALS

Ted JanssenGervais ChapuisMarc de Boissieu

Published in print: 2007-05-01

This chapter discusses aperiodic crystals. The structure and symmetry of quasiperiodic crystals can be described by embedding them into a higher-dimensional space as lattice periodic structures. Their intersection with the physical space gives the real structure, shifting the physical space parallel to the original one gives another possible realization of the crystal with the same energy. The Fourier transform of the quasiperiodic structure, and its diffraction pattern, are projections of the corresponding quantities in higher dimensions. The symmetry groups of quasiperiodic structures are superspace groups, higher-dimensional space groups for which the point group can be decomposed into a component in physical space and one in the additional, internal space. The structure determination reduces to the determination of number and positions of atomic surfaces in the higher-dimensional unit cell, and that of the shape of the atomic surfaces.

This chapter presents an analysis of experimental data which shows that the metal-insulator transition is possible in materials composed of atoms of only metallic elements. Such a transition may ...
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This chapter presents an analysis of experimental data which shows that the metal-insulator transition is possible in materials composed of atoms of only metallic elements. Such a transition may occur in spite of the high concentration of valence electrons. It requires stable atomic configurations to act as deep potential many-electron traps absorbing dozens of valence electrons. This means that bulk metallic space transforms into an assembly of quantum dots. Depending on the parameters, such a material either contains delocalized electrons (metal) or does not (insulator). Two types of substances with such properties are discussed: liquid binary intermetallic compounds and thermodynamically stable quasicrystals. The latter contain long-range order but do not have translational symmetry, Penrose tiling is a mathematical example.Less

CHEMICAL LOCALIZATION

V.F. Gantmakher

Published in print: 2005-08-25

This chapter presents an analysis of experimental data which shows that the metal-insulator transition is possible in materials composed of atoms of only metallic elements. Such a transition may occur in spite of the high concentration of valence electrons. It requires stable atomic configurations to act as deep potential many-electron traps absorbing dozens of valence electrons. This means that bulk metallic space transforms into an assembly of quantum dots. Depending on the parameters, such a material either contains delocalized electrons (metal) or does not (insulator). Two types of substances with such properties are discussed: liquid binary intermetallic compounds and thermodynamically stable quasicrystals. The latter contain long-range order but do not have translational symmetry, Penrose tiling is a mathematical example.

This book investigates different models of the fourth dimension and how these are applied in art and physics. It explores the distinction between the slicing, or Flatland, model and the projection, ...
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This book investigates different models of the fourth dimension and how these are applied in art and physics. It explores the distinction between the slicing, or Flatland, model and the projection, or shadow, model. The book compares the history of these two models and their uses and misuses in popular discussions. The book argues that Picasso used the projection model to invent cubism, and that Minkowski had four-dimensional projective geometry in mind when he structured special relativity. The discussion is brought to the present with an exposition of the projection model in the most creative ideas about space in contemporary mathematics such as twisters, quasicrystals, and quantum topology. The book proposes that the powerful role of projective geometry in the development of current mathematical ideas has been long overlooked and that our attachment to the slicing model is essentially a conceptual block that hinders progress in understanding contemporary models of spacetime.Less

Shadows of Reality : The Fourth Dimension in Relativity, Cubism, and Modern Thought

Tony Robbin

Published in print: 2006-03-31

This book investigates different models of the fourth dimension and how these are applied in art and physics. It explores the distinction between the slicing, or Flatland, model and the projection, or shadow, model. The book compares the history of these two models and their uses and misuses in popular discussions. The book argues that Picasso used the projection model to invent cubism, and that Minkowski had four-dimensional projective geometry in mind when he structured special relativity. The discussion is brought to the present with an exposition of the projection model in the most creative ideas about space in contemporary mathematics such as twisters, quasicrystals, and quantum topology. The book proposes that the powerful role of projective geometry in the development of current mathematical ideas has been long overlooked and that our attachment to the slicing model is essentially a conceptual block that hinders progress in understanding contemporary models of spacetime.

This chapter presents the basic properties of dynamical diffraction in an elementary way. The relationship with the band theory of solids is explained. The fundamental equations of dynamical theory ...
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This chapter presents the basic properties of dynamical diffraction in an elementary way. The relationship with the band theory of solids is explained. The fundamental equations of dynamical theory are given for scalar waves as a simplification; the solutions of the propagation equation are then derived for an incident plane wave in the 2-beam case; and the amplitude ratio between reflected and refracted waves deduced. The notions of wavefields, dispersion surface, and tie points are introduced. Two experimental set-ups are considered: transmission and reflection geometries. The boundary conditions at the entrance surface of the crystal are expressed in each case and the intensities of the refracted and reflected waves calculated as well as the anomalous absorption coefficient, due to the Borrmann effect, the Pendellösung interference fringe pattern and the integrated intensity. It is shown that the geometrical diffraction constitutes a limit of dynamical diffraction by small crystals. At the end of the chapter dynamic diffraction by quasicrystals is considered.Less

Elementary dynamical theory

ANDRÉ AUTHIER

Published in print: 2003-11-06

This chapter presents the basic properties of dynamical diffraction in an elementary way. The relationship with the band theory of solids is explained. The fundamental equations of dynamical theory are given for scalar waves as a simplification; the solutions of the propagation equation are then derived for an incident plane wave in the 2-beam case; and the amplitude ratio between reflected and refracted waves deduced. The notions of wavefields, dispersion surface, and tie points are introduced. Two experimental set-ups are considered: transmission and reflection geometries. The boundary conditions at the entrance surface of the crystal are expressed in each case and the intensities of the refracted and reflected waves calculated as well as the anomalous absorption coefficient, due to the Borrmann effect, the Pendellösung interference fringe pattern and the integrated intensity. It is shown that the geometrical diffraction constitutes a limit of dynamical diffraction by small crystals. At the end of the chapter dynamic diffraction by quasicrystals is considered.

The focus of this book is clearly on the statistics, topology, and geometry of crystal structures and crystal structure types. This allows one to uncover important structural relationships and to ...
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The focus of this book is clearly on the statistics, topology, and geometry of crystal structures and crystal structure types. This allows one to uncover important structural relationships and to illustrate the relative simplicity of most of the general structural building principles. It also allows one to show that a large variety of actual structures can be related to a rather small number of aristotypes. It is important that this book is readable and beneficial in the one way or another for everyone interested in intermetallic phases, from graduate students to experts in solid-state chemistry/physics/materials science. For that purpose it avoids using an enigmatic abstract terminology for the classification of structures. The focus on the statistical analysis of structures and structure types should be seen as an attempt to draw the background of the big picture of intermetallics, and to point to the white spots in it, which could be worthwhile exploring. This book was not planned as a textbook; rather, it should be a reference and guide through the incredibly rich world of intermetallic phases. In the first part of the book the basic concepts and tools are presented for the description of symmetry and structures of metallic elements and intermetallic phases, periodic and quasiperiodic ones, while in the second part the focus is on the discussion of their actual structures and properties.Less

Intermetallics : Structures, Properties, and Statistics

Walter SteurerJulia Dshemuchadse

Published in print: 2016-08-04

The focus of this book is clearly on the statistics, topology, and geometry of crystal structures and crystal structure types. This allows one to uncover important structural relationships and to illustrate the relative simplicity of most of the general structural building principles. It also allows one to show that a large variety of actual structures can be related to a rather small number of aristotypes. It is important that this book is readable and beneficial in the one way or another for everyone interested in intermetallic phases, from graduate students to experts in solid-state chemistry/physics/materials science. For that purpose it avoids using an enigmatic abstract terminology for the classification of structures. The focus on the statistical analysis of structures and structure types should be seen as an attempt to draw the background of the big picture of intermetallics, and to point to the white spots in it, which could be worthwhile exploring. This book was not planned as a textbook; rather, it should be a reference and guide through the incredibly rich world of intermetallic phases. In the first part of the book the basic concepts and tools are presented for the description of symmetry and structures of metallic elements and intermetallic phases, periodic and quasiperiodic ones, while in the second part the focus is on the discussion of their actual structures and properties.

Until the 1970s all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the following decades a new class of solid state matter, called aperiodic crystals, has been ...
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Until the 1970s all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the following decades a new class of solid state matter, called aperiodic crystals, has been found. It is a long-range ordered structure, but without lattice periodicity. It is found in a wide range of materials: organic and inorganic compounds, minerals (including a substantial portion of the earth’s crust), and metallic alloys, under various pressures and temperatures. Because of the lack of periodicity the usual techniques for the study of structure and physical properties no longer work, and new techniques have to be developed. This book deals with the characterization of the structure, the structure determination, and the study of the physical properties, especially the dynamical and electronic properties of aperiodic crystals. The treatment is based on a description in a space with more dimensions than three, the so-called superspace. This allows us to generalize the standard crystallography and to look differently at the dynamics. The three main classes of aperiodic crystals, modulated phases, incommensurate composites, and quasicrystals are treated from a unified point of view which stresses the similarities of the various systems. The book assumes as a prerequisite a knowledge of the fundamental techniques of crystallography and the theory of condensed matter, and covers the literature at the forefront of the field.Less

Ted JanssenGervais ChapuisMarc de Boissieu

Published in print: 2018-06-07

Until the 1970s all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the following decades a new class of solid state matter, called aperiodic crystals, has been found. It is a long-range ordered structure, but without lattice periodicity. It is found in a wide range of materials: organic and inorganic compounds, minerals (including a substantial portion of the earth’s crust), and metallic alloys, under various pressures and temperatures. Because of the lack of periodicity the usual techniques for the study of structure and physical properties no longer work, and new techniques have to be developed. This book deals with the characterization of the structure, the structure determination, and the study of the physical properties, especially the dynamical and electronic properties of aperiodic crystals. The treatment is based on a description in a space with more dimensions than three, the so-called superspace. This allows us to generalize the standard crystallography and to look differently at the dynamics. The three main classes of aperiodic crystals, modulated phases, incommensurate composites, and quasicrystals are treated from a unified point of view which stresses the similarities of the various systems. The book assumes as a prerequisite a knowledge of the fundamental techniques of crystallography and the theory of condensed matter, and covers the literature at the forefront of the field.

In this chapter, the two main classes of intermetallic QCs known so far are introduced: decagonal QCs and icosahedral QCs. The terminology “decagonal” and “icosahedral”, respectively, refers to the ...
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In this chapter, the two main classes of intermetallic QCs known so far are introduced: decagonal QCs and icosahedral QCs. The terminology “decagonal” and “icosahedral”, respectively, refers to the Laue symmetry (10/m, 10/mmm and m3¯5¯, respectively) of their diffraction patterns (intensity weighted reciprocal lattice) or, equivalently, to the symmetry of the interatomic vector map (auto-correlation function or Patterson map). It also refers to the “bond-orientational order” of a QC structure, what is nothing else but its vector map. The full spacegroup symmetry of a quasiperiodic structure can best be described in the framework of the nD approach. However, an equivalent description is also possible in 3D reciprocal space based on the symmetry relationships between the complex structure factors.Less

Quasicrystals (QCs)

Walter SteurerJulia Dshemuchadse

Published in print: 2016-08-04

In this chapter, the two main classes of intermetallic QCs known so far are introduced: decagonal QCs and icosahedral QCs. The terminology “decagonal” and “icosahedral”, respectively, refers to the Laue symmetry (10/m, 10/mmm and m3¯5¯, respectively) of their diffraction patterns (intensity weighted reciprocal lattice) or, equivalently, to the symmetry of the interatomic vector map (auto-correlation function or Patterson map). It also refers to the “bond-orientational order” of a QC structure, what is nothing else but its vector map. The full spacegroup symmetry of a quasiperiodic structure can best be described in the framework of the nD approach. However, an equivalent description is also possible in 3D reciprocal space based on the symmetry relationships between the complex structure factors.

This chapter focuses on patterns; these were previously believed to be regularly repeating motifs. It was only in 1964 when Robert Berger discovered that nonrepeating patterns actually existed. The ...
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This chapter focuses on patterns; these were previously believed to be regularly repeating motifs. It was only in 1964 when Robert Berger discovered that nonrepeating patterns actually existed. The discussion looks at how tiles could be arranged in a certain way so as not to repeat yet still form a pattern, which culminated in Nicolaas de Bruijn's repeating system that led to a perfect nonrepeating pattern. This system involves the projection method and the dual method, and these are found to be essentially the same. The chapter finishes by looking at quasicrystals and three main ways to make a tessellation.Less

Patterns, Crystals, and Projections

Tony Robbin

Published in print: 2006-03-31

This chapter focuses on patterns; these were previously believed to be regularly repeating motifs. It was only in 1964 when Robert Berger discovered that nonrepeating patterns actually existed. The discussion looks at how tiles could be arranged in a certain way so as not to repeat yet still form a pattern, which culminated in Nicolaas de Bruijn's repeating system that led to a perfect nonrepeating pattern. This system involves the projection method and the dual method, and these are found to be essentially the same. The chapter finishes by looking at quasicrystals and three main ways to make a tessellation.

The origin of the stability of aperiodic systems is very difficult to answer. Often the terms ‘competitive forces’ or ‘frustration’ have been proposed as the origin of stability. The role of Fermi ...
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The origin of the stability of aperiodic systems is very difficult to answer. Often the terms ‘competitive forces’ or ‘frustration’ have been proposed as the origin of stability. The role of Fermi surfaces and Brillouin zone boundary have also been invoked. This chapter deals with the numerous attempts which have been proposed for a better understanding. First, the Landau theory of phase transition, which has often been applied to understand the stability of incommensurate and composite systems, is presented here. Various semi-microscopic models are also proposed, in particular the Frenkel–Kontorova and Frank–Van der Merwe models, as well as spin models. Phase diagrams have been calculated with some success with the ANNI and DIFFOUR models. For quasicrystals, only the simplest general features are found in model systems. For a better understanding, more complex calculations are required, using, for example, ab initio methods. The chapter also discusses electronic instabilities, charge-density systems, Hume–Rothery compounds, and the growth of quasicrystals.Less

Origin and stability

Ted JanssenGervais ChapuisMarc de Boissieu

Published in print: 2018-06-07

The origin of the stability of aperiodic systems is very difficult to answer. Often the terms ‘competitive forces’ or ‘frustration’ have been proposed as the origin of stability. The role of Fermi surfaces and Brillouin zone boundary have also been invoked. This chapter deals with the numerous attempts which have been proposed for a better understanding. First, the Landau theory of phase transition, which has often been applied to understand the stability of incommensurate and composite systems, is presented here. Various semi-microscopic models are also proposed, in particular the Frenkel–Kontorova and Frank–Van der Merwe models, as well as spin models. Phase diagrams have been calculated with some success with the ANNI and DIFFOUR models. For quasicrystals, only the simplest general features are found in model systems. For a better understanding, more complex calculations are required, using, for example, ab initio methods. The chapter also discusses electronic instabilities, charge-density systems, Hume–Rothery compounds, and the growth of quasicrystals.

The study of symmetry elements in two and three dimensions is followed by point groups, their derivation and recognition, including an interactive program for point group recognition. Euler’s theorem ...
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The study of symmetry elements in two and three dimensions is followed by point groups, their derivation and recognition, including an interactive program for point group recognition. Euler’s theorem on the combination of rotations is discussed, and the physical properties of crystals and molecules in relation to their point groups explained. Chemical examples of crystallographic and non-crystallographic point groups illustrated by stereoscopic drawings. The discussion is extended to quasicrystals, and to fivefold symmetry in crystalline material.Less

Point group symmetry

Mark Ladd

Published in print: 2014-02-20

The study of symmetry elements in two and three dimensions is followed by point groups, their derivation and recognition, including an interactive program for point group recognition. Euler’s theorem on the combination of rotations is discussed, and the physical properties of crystals and molecules in relation to their point groups explained. Chemical examples of crystallographic and non-crystallographic point groups illustrated by stereoscopic drawings. The discussion is extended to quasicrystals, and to fivefold symmetry in crystalline material.